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A320067
Expansion of Product_{k>0} theta_3(q^k), where theta_3() is the Jacobi theta function.
21
1, 2, 2, 6, 8, 10, 22, 26, 36, 60, 78, 106, 152, 202, 258, 370, 478, 602, 828, 1042, 1332, 1758, 2198, 2758, 3572, 4448, 5518, 7012, 8636, 10654, 13350, 16362, 19946, 24722, 30070, 36478, 44776, 54010, 65202, 79234, 95196, 114166, 137686, 164530, 196252, 235308, 279718, 332002
OFFSET
0,2
COMMENTS
Also the number of integer solutions (a_1, a_2, ..., a_n) to the equation a_1^2 + 2*a_2^2 + ... + n*a_n^2 = n.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)
Eric Weisstein's World of Mathematics, Jacobi Theta Functions
FORMULA
Expansion of Product_{k>0} eta(q^(2*k))^5 / (eta(q^k)*eta(q^(4*k)))^2.
a(n) ~ log(2)^(3/8) * exp(Pi*sqrt(n*log(2))) / (4 * Pi^(1/4) * n^(7/8)). - Vaclav Kotesovec, Oct 05 2018
Expansion of Product_{k>0} theta_4(q^(2*k))/theta_4(q^(2*k-1)), where theta_4() is the Jacobi theta function. - Seiichi Manyama, Oct 26 2018
MATHEMATICA
nmax = 50; CoefficientList[Series[Product[EllipticTheta[3, 0, x^k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 05 2018 *)
nmax = 50; CoefficientList[Series[Product[(1 - x^(k*j))*(1 + x^(k*j))^3/(1 + x^(2*k*j))^2, {k, 1, nmax}, {j, 1, Floor[nmax/k] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 05 2018 *)
PROG
(PARI) m=50; x='x+O('x^m); Vec(1/(prod(k=1, 2*m, prod(j=1, floor(2*m/k), (1 - x^(k*j))*(1 + x^(k*j))^3/(1 + x^(2*k*j))^2 )))) \\ G. C. Greubel, Oct 29 2018
(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[(&*[(1 - x^(k*j))*(1 + x^(k*j))^3/(1 + x^(2*k*j))^2: j in [1..Floor(2*m/k)]]): k in [1..2*m]]))); // G. C. Greubel, Oct 29 2018
KEYWORD
nonn,nice
AUTHOR
Seiichi Manyama, Oct 05 2018
STATUS
approved